11.12.2: General Forms and Special Forms

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In any crystal class, a general form, which we generically designate {hkl}, contains faces that intersect the three crystal axes at different distances from the origin. In the cubic system, the Miller indices of the general form must all be different integers because unit cell dimensions are the same for all three axes (see Box 11-4). For other systems, where some unit cell dimension are the same and some are different, two or more Miller indices may be the same. Thus, in the orthorhombic, monoclinic and triclinic systems the general form can be symbolized {hkl}, but {111} or {hhh} are also general forms. (We use three h’s in the symbol if we wish to show that all indices have the same integer value.) In the tetragonal system, the first two indices of the general form must be different since the a and b unit cell dimensions are the same. Thus, we may designate a tetragonal general form, for example, as {121} or {122}. In the hexagonal system, all three indices must be different and the general form is {hkl} or {123}, equivalent to any other form with three different Miller index values. In any crystal class, the general form contains the maximum number of faces that can be related by class symmetry.

11.69.png — Figure 11.69: Some orthorhombic forms in crystal class ²/_m²/_m²/_m

In contrast with general forms, special forms are parallel to one or two crystal axes. This means they are parallel or perpendicular to rotation axes or mirror planes. Consequently, special forms contain fewer faces than general forms for the same crystal class. For example, the drawing on the left in Figure 11.69 shows the general form for crystal class ²/_m²/_m²/_m. It is an orthorhombic dipyramid. It contains eight faces, four of one shape and four of another. The middle drawing shows a crystal that contains six faces and two special forms, an orthorhombic pinacoid (top and bottom faces) and an orthorhombic prism (four vertical faces). The third crystal contains all three forms: the general form and the two special forms. All three crystals have the same symmetry.

General Forms, Special Forms, and Miller Indices

The relationship between special forms and general forms is the same as the one between special points and general points discussed in the previous chapter. Faces of a general form are neither parallel nor perpendicular to any symmetry element, while faces of special forms are. Thus, in special forms, symmetry relates fewer equivalent faces.

11.70.png — Figure 11.70: Hexoctahedron and related special forms

The hexoctahedron (Figure 11.70) is the general form in the ⁴/_m3²/_m point group; its Miller index contains three different values. The simplest index is {123}, indicating that h ≠ k ≠ l and that none of the indices are zero. The hexoctahedron contains 48 triangular faces. We can think of this form as octahedron with each of its eight faces replaced by six smaller triangular faces; hence, its name.

All other forms with the same symmetry are special forms in the hexoctahedral class. Crystals in this class may contain any of 15 forms singly or in combinations. Figure 11.70 shows four examples. See the table in Chapter 10 for other examples.

In the cubic system, special forms have two or more identical indices in their Miller index, or have an index with value zero; Figure 11.70 contains examples. For example cube faces belong to the {h00} form; octahedron faces belong to the {hhh} form. The dodecahedron faces belong to the {hh0} form. Another related form, the trapezohedron, has Miller index {hkk}, signifying that two of its indices have the same value and that none are zero. h, k, and l can have any integer values and these relationships will hold.

As shown in Figure 11.70, if we combine groups of faces on a hexoctahedron and replace them with a single face, we get a cube, octahedron, dodecahedron, or trapezohedron with no change in symmetry.

Faces of the special forms all coincide with symmetry elements – that is what makes them special. Cube faces are perpendicular to 4-fold rotation axes, octahedron faces are perpendicular to 3-fold rotoinversion axes, dodecahedron faces are perpendicular to 2-fold axes, and trapezohedron faces are perpendicular to mirror planes.

The examples seen in Figure 11.70 are all from a single point group, ⁴/_m3²/_m. But the principles discussed apply to all 32 point groups. However, symmetry relationships and the nature of special forms are more difficult to see for some point groups, especially those with little symmetry.